Solving quadratic equations using reduced unimodular quadratic forms
Reads0
Chats0
TLDR
A generalized LLL algorithm to reduce the quadratic form of an n x n symmetric matrix with integral entries and with det Q ¬= 0 is described, which is proved to run in polynomial time.Abstract:
Let Q be an n x n symmetric matrix with integral entries and with det Q ¬= 0, but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to run in polynomial time. We also describe an algorithm for the minimization of a ternary quadratic form: when a quadratic equation q(x,y,z) = 0 is solvable over Q, a solution can be deduced from another quadratic equation of determinant ±1. The combination of these algorithms allows us to solve efficiently any general ternary quadratic equation over Q, and this gives a polynomial time algorithm (as soon as the factorization of the determinant of Q is known).read more
Citations
More filters
Journal ArticleDOI
An LLL Algorithm with Quadratic Complexity
Phong Q. Nguyen,Damien Stehlé +1 more
TL;DR: The ${\rm L}^2$ algorithm is introduced, a new and natural floating-point variant of the Lenstra-Lenstra-Lovasz lattice basis reduction algorithm which provably outputs £3,000-reduced bases in polynomial time and is the first algorithm whose running time provably grows only quadratically with respect to${\rm log}\,B$, like Euclid's gcd algorithm and Lagrange's two-dimensional algorithm.
Journal ArticleDOI
Efficient decomposition of single-qubit gates intoVbasis circuits
TL;DR: The first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis are developed, an alternative universal basis to the more commonly studied $\{H,T\}$ basis.
Journal ArticleDOI
Classification of nilpotent associative algebras of small dimension
TL;DR: It is shown that nilpotent associative algebras of dimensions up to 4 over any field can be classified as central extensions of algeBRas of smaller dimension, analogous to methods known fornilpotent Lie algebraes.
Book ChapterDOI
Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms
TL;DR: In this article, the relationship between quaternion algebras and quadratic forms with a focus on computational aspects is discussed and the basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2-matrix ring M2(R) and, if so, to compute such an embedding.
References
More filters
Book
A Course in Computational Algebraic Number Theory
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Book
Rational Quadratic Forms
TL;DR: In this article, the authors present local-global theorems and their limitations, that is, the information that can be obtained from p-adic considerations, and some of the results extend to algebraic number fields or in other ways.
Book
The Algorithmic Resolution of Diophantine Equations
TL;DR: In this article, the LLL-algorithm was used to solve diophantine equations with linear forms in logarithms and rational points on elliptic curves, respectively.
Journal ArticleDOI
Efficient solution of rational conics
John Cremona,David J. Rusin +1 more
TL;DR: This work presents efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions and shows that no integer factorization is required.